Let $G$ be a finite group and $\mathcal{L}(G)$ its subgroup lattice. Let $\mu$ be the Möbius function on $\mathcal{L}(G)$.

The reduced Euler characteristic of the order complex of the coset poset $\{ Kg \ | \ K<G, \ g \in G \} $ is $$\chi(G) := -\sum_{H \in \mathcal{L}(G)} \mu(H,G)|G:H|.$$ Gaschütz showed that $\chi(G)$ is nonzero if $G$ is solvable and the question whether $\chi(G)$ is nonzero for any finite group $G$ is an open problem motivated by Brown (see DOI: 10.1016/j.aim.2015.10.018).

There is a relative generalization of this problem. Let $H$ be a subgroup of $G$. The reduced Euler characteristic of the order complex of the coset poset $\{ Kg \ | \ K \in [H,G), \ g \in G \} $ is $$\chi(H,G) := -\sum_{K \in [H,G]} \mu(K,G)|G:K|.$$
The question whether $\chi(H,G)$ is nonzero for any interval of finite groups $[H,G]$ is also open.